Theorem caubl 23106

Description: Sufficient condition to ensure a sequence of nested balls is Cauchy. (Contributed by Mario Carneiro, 18-Jan-2014.) (Revised by Mario Carneiro, 1-May-2014.)

Hypotheses

Ref	Expression
caubl.2	|- ( ph -> D e. ( *Met ` X ) )
caubl.3	|- ( ph -> F : NN --> ( X X. RR+ ) )
caubl.4	|- ( ph -> A. n e. NN ( ( ball ` D ) ` ( F ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) )
caubl.5	|- ( ph -> A. r e. RR+ E. n e. NN ( 2nd ` ( F ` n ) ) < r )

Assertion

Ref	Expression
caubl	|- ( ph -> ( 1st o. F ) e. ( Cau ` D ) )

Distinct variable groups:   n, r, D   n, F, r   ph, r   n, X, r   ph, n

Proof of Theorem caubl

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caubl.5	. . 3 |- ( ph -> A. r e. RR+ E. n e. NN ( 2nd ` ( F ` n ) ) < r )
2		fveq2 6191	. . . . . . . . . . . . . 14 |- ( r = n -> ( F ` r ) = ( F ` n ) )
3	2	fveq2d 6195	. . . . . . . . . . . . 13 |- ( r = n -> ( ( ball ` D ) ` ( F ` r ) ) = ( ( ball ` D ) ` ( F ` n ) ) )
4	3	sseq1d 3632	. . . . . . . . . . . 12 |- ( r = n -> ( ( ( ball ` D ) ` ( F ` r ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) <-> ( ( ball ` D ) ` ( F ` n ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) ) )
5	4	imbi2d 330	. . . . . . . . . . 11 |- ( r = n -> ( ( ( ph /\ n e. NN ) -> ( ( ball ` D ) ` ( F ` r ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) ) <-> ( ( ph /\ n e. NN ) -> ( ( ball ` D ) ` ( F ` n ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) ) ) )
6		fveq2 6191	. . . . . . . . . . . . . 14 |- ( r = k -> ( F ` r ) = ( F ` k ) )
7	6	fveq2d 6195	. . . . . . . . . . . . 13 |- ( r = k -> ( ( ball ` D ) ` ( F ` r ) ) = ( ( ball ` D ) ` ( F ` k ) ) )
8	7	sseq1d 3632	. . . . . . . . . . . 12 |- ( r = k -> ( ( ( ball ` D ) ` ( F ` r ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) <-> ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) ) )
9	8	imbi2d 330	. . . . . . . . . . 11 |- ( r = k -> ( ( ( ph /\ n e. NN ) -> ( ( ball ` D ) ` ( F ` r ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) ) <-> ( ( ph /\ n e. NN ) -> ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) ) ) )
10		fveq2 6191	. . . . . . . . . . . . . 14 |- ( r = ( k + 1 ) -> ( F ` r ) = ( F ` ( k + 1 ) ) )
11	10	fveq2d 6195	. . . . . . . . . . . . 13 |- ( r = ( k + 1 ) -> ( ( ball ` D ) ` ( F ` r ) ) = ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) )
12	11	sseq1d 3632	. . . . . . . . . . . 12 |- ( r = ( k + 1 ) -> ( ( ( ball ` D ) ` ( F ` r ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) <-> ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) ) )
13	12	imbi2d 330	. . . . . . . . . . 11 |- ( r = ( k + 1 ) -> ( ( ( ph /\ n e. NN ) -> ( ( ball ` D ) ` ( F ` r ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) ) <-> ( ( ph /\ n e. NN ) -> ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) ) ) )
14		ssid 3624	. . . . . . . . . . . 12 |- ( ( ball ` D ) ` ( F ` n ) ) C_ ( ( ball ` D ) ` ( F ` n ) )
15	14	2a1i 12	. . . . . . . . . . 11 |- ( n e. ZZ -> ( ( ph /\ n e. NN ) -> ( ( ball ` D ) ` ( F ` n ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) ) )
16		caubl.4	. . . . . . . . . . . . . . . 16 |- ( ph -> A. n e. NN ( ( ball ` D ) ` ( F ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) )
17		eluznn 11758	. . . . . . . . . . . . . . . 16 |- ( ( n e. NN /\ k e. ( ZZ>= ` n ) ) -> k e. NN )
18		oveq1 6657	. . . . . . . . . . . . . . . . . . . 20 |- ( n = k -> ( n + 1 ) = ( k + 1 ) )
19	18	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 |- ( n = k -> ( F ` ( n + 1 ) ) = ( F ` ( k + 1 ) ) )
20	19	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 |- ( n = k -> ( ( ball ` D ) ` ( F ` ( n + 1 ) ) ) = ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) )
21		fveq2 6191	. . . . . . . . . . . . . . . . . . 19 |- ( n = k -> ( F ` n ) = ( F ` k ) )
22	21	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 |- ( n = k -> ( ( ball ` D ) ` ( F ` n ) ) = ( ( ball ` D ) ` ( F ` k ) ) )
23	20, 22	sseq12d 3634	. . . . . . . . . . . . . . . . 17 |- ( n = k -> ( ( ( ball ` D ) ` ( F ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) <-> ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` k ) ) ) )
24	23	rspccva 3308	. . . . . . . . . . . . . . . 16 |- ( ( A. n e. NN ( ( ball ` D ) ` ( F ` ( n + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) /\ k e. NN ) -> ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` k ) ) )
25	16, 17, 24	syl2an 494	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( n e. NN /\ k e. ( ZZ>= ` n ) ) ) -> ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` k ) ) )
26	25	anassrs 680	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ n e. NN ) /\ k e. ( ZZ>= ` n ) ) -> ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` k ) ) )
27		sstr2 3610	. . . . . . . . . . . . . 14 |- ( ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` k ) ) -> ( ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) -> ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) ) )
28	26, 27	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. NN ) /\ k e. ( ZZ>= ` n ) ) -> ( ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) -> ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) ) )
29	28	expcom 451	. . . . . . . . . . . 12 |- ( k e. ( ZZ>= ` n ) -> ( ( ph /\ n e. NN ) -> ( ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) -> ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) ) ) )
30	29	a2d 29	. . . . . . . . . . 11 |- ( k e. ( ZZ>= ` n ) -> ( ( ( ph /\ n e. NN ) -> ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) ) -> ( ( ph /\ n e. NN ) -> ( ( ball ` D ) ` ( F ` ( k + 1 ) ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) ) ) )
31	5, 9, 13, 9, 15, 30	uzind4 11746	. . . . . . . . . 10 |- ( k e. ( ZZ>= ` n ) -> ( ( ph /\ n e. NN ) -> ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) ) )
32	31	com12 32	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( k e. ( ZZ>= ` n ) -> ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) ) )
33	32	ad2ant2r 783	. . . . . . . 8 |- ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) -> ( k e. ( ZZ>= ` n ) -> ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) ) )
34		relxp 5227	. . . . . . . . . . . . . . . 16 |- Rel ( X X. RR+ )
35		caubl.3	. . . . . . . . . . . . . . . . . 18 |- ( ph -> F : NN --> ( X X. RR+ ) )
36	35	ad3antrrr 766	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> F : NN --> ( X X. RR+ ) )
37		simplrl 800	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> n e. NN )
38	36, 37	ffvelrnd 6360	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( F ` n ) e. ( X X. RR+ ) )
39		1st2nd 7214	. . . . . . . . . . . . . . . 16 |- ( ( Rel ( X X. RR+ ) /\ ( F ` n ) e. ( X X. RR+ ) ) -> ( F ` n ) = <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. )
40	34, 38, 39	sylancr 695	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( F ` n ) = <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. )
41	40	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( ( ball ` D ) ` ( F ` n ) ) = ( ( ball ` D ) ` <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. ) )
42		df-ov 6653	. . . . . . . . . . . . . 14 |- ( ( 1st ` ( F ` n ) ) ( ball ` D ) ( 2nd ` ( F ` n ) ) ) = ( ( ball ` D ) ` <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. )
43	41, 42	syl6eqr 2674	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( ( ball ` D ) ` ( F ` n ) ) = ( ( 1st ` ( F ` n ) ) ( ball ` D ) ( 2nd ` ( F ` n ) ) ) )
44		caubl.2	. . . . . . . . . . . . . . 15 |- ( ph -> D e. ( *Met ` X ) )
45	44	ad3antrrr 766	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> D e. ( *Met ` X ) )
46		xp1st 7198	. . . . . . . . . . . . . . 15 |- ( ( F ` n ) e. ( X X. RR+ ) -> ( 1st ` ( F ` n ) ) e. X )
47	38, 46	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( 1st ` ( F ` n ) ) e. X )
48		xp2nd 7199	. . . . . . . . . . . . . . . 16 |- ( ( F ` n ) e. ( X X. RR+ ) -> ( 2nd ` ( F ` n ) ) e. RR+ )
49	38, 48	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( 2nd ` ( F ` n ) ) e. RR+ )
50	49	rpxrd 11873	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( 2nd ` ( F ` n ) ) e. RR* )
51		simpllr 799	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> r e. RR+ )
52	51	rpxrd 11873	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> r e. RR* )
53		simplrr 801	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( 2nd ` ( F ` n ) ) < r )
54		rpre 11839	. . . . . . . . . . . . . . . . 17 |- ( ( 2nd ` ( F ` n ) ) e. RR+ -> ( 2nd ` ( F ` n ) ) e. RR )
55		rpre 11839	. . . . . . . . . . . . . . . . 17 |- ( r e. RR+ -> r e. RR )
56		ltle 10126	. . . . . . . . . . . . . . . . 17 |- ( ( ( 2nd ` ( F ` n ) ) e. RR /\ r e. RR ) -> ( ( 2nd ` ( F ` n ) ) < r -> ( 2nd ` ( F ` n ) ) <_ r ) )
57	54, 55, 56	syl2an 494	. . . . . . . . . . . . . . . 16 |- ( ( ( 2nd ` ( F ` n ) ) e. RR+ /\ r e. RR+ ) -> ( ( 2nd ` ( F ` n ) ) < r -> ( 2nd ` ( F ` n ) ) <_ r ) )
58	49, 51, 57	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( ( 2nd ` ( F ` n ) ) < r -> ( 2nd ` ( F ` n ) ) <_ r ) )
59	53, 58	mpd 15	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( 2nd ` ( F ` n ) ) <_ r )
60		ssbl 22228	. . . . . . . . . . . . . 14 |- ( ( ( D e. ( *Met ` X ) /\ ( 1st ` ( F ` n ) ) e. X ) /\ ( ( 2nd ` ( F ` n ) ) e. RR* /\ r e. RR* ) /\ ( 2nd ` ( F ` n ) ) <_ r ) -> ( ( 1st ` ( F ` n ) ) ( ball ` D ) ( 2nd ` ( F ` n ) ) ) C_ ( ( 1st ` ( F ` n ) ) ( ball ` D ) r ) )
61	45, 47, 50, 52, 59, 60	syl221anc 1337	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( ( 1st ` ( F ` n ) ) ( ball ` D ) ( 2nd ` ( F ` n ) ) ) C_ ( ( 1st ` ( F ` n ) ) ( ball ` D ) r ) )
62	43, 61	eqsstrd 3639	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( ( ball ` D ) ` ( F ` n ) ) C_ ( ( 1st ` ( F ` n ) ) ( ball ` D ) r ) )
63		sstr2 3610	. . . . . . . . . . . 12 |- ( ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) -> ( ( ( ball ` D ) ` ( F ` n ) ) C_ ( ( 1st ` ( F ` n ) ) ( ball ` D ) r ) -> ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( 1st ` ( F ` n ) ) ( ball ` D ) r ) ) )
64	62, 63	syl5com 31	. . . . . . . . . . 11 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) -> ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( 1st ` ( F ` n ) ) ( ball ` D ) r ) ) )
65		simprl 794	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) -> n e. NN )
66	65, 17	sylan 488	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> k e. NN )
67	36, 66	ffvelrnd 6360	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( F ` k ) e. ( X X. RR+ ) )
68		xp1st 7198	. . . . . . . . . . . . . 14 |- ( ( F ` k ) e. ( X X. RR+ ) -> ( 1st ` ( F ` k ) ) e. X )
69	67, 68	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( 1st ` ( F ` k ) ) e. X )
70		xp2nd 7199	. . . . . . . . . . . . . 14 |- ( ( F ` k ) e. ( X X. RR+ ) -> ( 2nd ` ( F ` k ) ) e. RR+ )
71	67, 70	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( 2nd ` ( F ` k ) ) e. RR+ )
72		blcntr 22218	. . . . . . . . . . . . 13 |- ( ( D e. ( *Met ` X ) /\ ( 1st ` ( F ` k ) ) e. X /\ ( 2nd ` ( F ` k ) ) e. RR+ ) -> ( 1st ` ( F ` k ) ) e. ( ( 1st ` ( F ` k ) ) ( ball ` D ) ( 2nd ` ( F ` k ) ) ) )
73	45, 69, 71, 72	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( 1st ` ( F ` k ) ) e. ( ( 1st ` ( F ` k ) ) ( ball ` D ) ( 2nd ` ( F ` k ) ) ) )
74		1st2nd 7214	. . . . . . . . . . . . . . 15 |- ( ( Rel ( X X. RR+ ) /\ ( F ` k ) e. ( X X. RR+ ) ) -> ( F ` k ) = <. ( 1st ` ( F ` k ) ) , ( 2nd ` ( F ` k ) ) >. )
75	34, 67, 74	sylancr 695	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( F ` k ) = <. ( 1st ` ( F ` k ) ) , ( 2nd ` ( F ` k ) ) >. )
76	75	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( ( ball ` D ) ` ( F ` k ) ) = ( ( ball ` D ) ` <. ( 1st ` ( F ` k ) ) , ( 2nd ` ( F ` k ) ) >. ) )
77		df-ov 6653	. . . . . . . . . . . . 13 |- ( ( 1st ` ( F ` k ) ) ( ball ` D ) ( 2nd ` ( F ` k ) ) ) = ( ( ball ` D ) ` <. ( 1st ` ( F ` k ) ) , ( 2nd ` ( F ` k ) ) >. )
78	76, 77	syl6eqr 2674	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( ( ball ` D ) ` ( F ` k ) ) = ( ( 1st ` ( F ` k ) ) ( ball ` D ) ( 2nd ` ( F ` k ) ) ) )
79	73, 78	eleqtrrd 2704	. . . . . . . . . . 11 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( 1st ` ( F ` k ) ) e. ( ( ball ` D ) ` ( F ` k ) ) )
80		ssel 3597	. . . . . . . . . . 11 |- ( ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( 1st ` ( F ` n ) ) ( ball ` D ) r ) -> ( ( 1st ` ( F ` k ) ) e. ( ( ball ` D ) ` ( F ` k ) ) -> ( 1st ` ( F ` k ) ) e. ( ( 1st ` ( F ` n ) ) ( ball ` D ) r ) ) )
81	64, 79, 80	syl6ci 71	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) -> ( 1st ` ( F ` k ) ) e. ( ( 1st ` ( F ` n ) ) ( ball ` D ) r ) ) )
82		elbl2 22195	. . . . . . . . . . 11 |- ( ( ( D e. ( *Met ` X ) /\ r e. RR* ) /\ ( ( 1st ` ( F ` n ) ) e. X /\ ( 1st ` ( F ` k ) ) e. X ) ) -> ( ( 1st ` ( F ` k ) ) e. ( ( 1st ` ( F ` n ) ) ( ball ` D ) r ) <-> ( ( 1st ` ( F ` n ) ) D ( 1st ` ( F ` k ) ) ) < r ) )
83	45, 52, 47, 69, 82	syl22anc 1327	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( ( 1st ` ( F ` k ) ) e. ( ( 1st ` ( F ` n ) ) ( ball ` D ) r ) <-> ( ( 1st ` ( F ` n ) ) D ( 1st ` ( F ` k ) ) ) < r ) )
84	81, 83	sylibd 229	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) /\ k e. ( ZZ>= ` n ) ) -> ( ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) -> ( ( 1st ` ( F ` n ) ) D ( 1st ` ( F ` k ) ) ) < r ) )
85	84	ex 450	. . . . . . . 8 |- ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) -> ( k e. ( ZZ>= ` n ) -> ( ( ( ball ` D ) ` ( F ` k ) ) C_ ( ( ball ` D ) ` ( F ` n ) ) -> ( ( 1st ` ( F ` n ) ) D ( 1st ` ( F ` k ) ) ) < r ) ) )
86	33, 85	mpdd 43	. . . . . . 7 |- ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) -> ( k e. ( ZZ>= ` n ) -> ( ( 1st ` ( F ` n ) ) D ( 1st ` ( F ` k ) ) ) < r ) )
87	86	ralrimiv 2965	. . . . . 6 |- ( ( ( ph /\ r e. RR+ ) /\ ( n e. NN /\ ( 2nd ` ( F ` n ) ) < r ) ) -> A. k e. ( ZZ>= ` n ) ( ( 1st ` ( F ` n ) ) D ( 1st ` ( F ` k ) ) ) < r )
88	87	expr 643	. . . . 5 |- ( ( ( ph /\ r e. RR+ ) /\ n e. NN ) -> ( ( 2nd ` ( F ` n ) ) < r -> A. k e. ( ZZ>= ` n ) ( ( 1st ` ( F ` n ) ) D ( 1st ` ( F ` k ) ) ) < r ) )
89	88	reximdva 3017	. . . 4 |- ( ( ph /\ r e. RR+ ) -> ( E. n e. NN ( 2nd ` ( F ` n ) ) < r -> E. n e. NN A. k e. ( ZZ>= ` n ) ( ( 1st ` ( F ` n ) ) D ( 1st ` ( F ` k ) ) ) < r ) )
90	89	ralimdva 2962	. . 3 |- ( ph -> ( A. r e. RR+ E. n e. NN ( 2nd ` ( F ` n ) ) < r -> A. r e. RR+ E. n e. NN A. k e. ( ZZ>= ` n ) ( ( 1st ` ( F ` n ) ) D ( 1st ` ( F ` k ) ) ) < r ) )
91	1, 90	mpd 15	. 2 |- ( ph -> A. r e. RR+ E. n e. NN A. k e. ( ZZ>= ` n ) ( ( 1st ` ( F ` n ) ) D ( 1st ` ( F ` k ) ) ) < r )
92		nnuz 11723	. . 3 |- NN = ( ZZ>= ` 1 )
93		1zzd 11408	. . 3 |- ( ph -> 1 e. ZZ )
94		fvco3 6275	. . . 4 |- ( ( F : NN --> ( X X. RR+ ) /\ k e. NN ) -> ( ( 1st o. F ) ` k ) = ( 1st ` ( F ` k ) ) )
95	35, 94	sylan 488	. . 3 |- ( ( ph /\ k e. NN ) -> ( ( 1st o. F ) ` k ) = ( 1st ` ( F ` k ) ) )
96		fvco3 6275	. . . 4 |- ( ( F : NN --> ( X X. RR+ ) /\ n e. NN ) -> ( ( 1st o. F ) ` n ) = ( 1st ` ( F ` n ) ) )
97	35, 96	sylan 488	. . 3 |- ( ( ph /\ n e. NN ) -> ( ( 1st o. F ) ` n ) = ( 1st ` ( F ` n ) ) )
98		1stcof 7196	. . . 4 |- ( F : NN --> ( X X. RR+ ) -> ( 1st o. F ) : NN --> X )
99	35, 98	syl 17	. . 3 |- ( ph -> ( 1st o. F ) : NN --> X )
100	92, 44, 93, 95, 97, 99	iscauf 23078	. 2 |- ( ph -> ( ( 1st o. F ) e. ( Cau ` D ) <-> A. r e. RR+ E. n e. NN A. k e. ( ZZ>= ` n ) ( ( 1st ` ( F ` n ) ) D ( 1st ` ( F ` k ) ) ) < r ) )
101	91, 100	mpbird 247	1 |- ( ph -> ( 1st o. F ) e. ( Cau ` D ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   <.cop 4183   class class class wbr 4653   X. cxp 5112   o. ccom 5118   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   RRcr 9935   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   *Metcxmt 19731   ballcbl 19733   Caucca 23051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-psmet 19738  df-xmet 19739  df-bl 19741  df-cau 23054

This theorem is referenced by:  bcthlem4  23124  heiborlem9  33618